Summary: Circular Bernstein­B'ezier Polynomials
Peter Alfeld, Marian Neamtu, and Larry L. Schumaker
Abstract. In this paper we discuss a natural way to define barycen­
tric coordinates associated with circular arcs. This leads to a theory of
Bernstein­B'ezier polynomials which parallels the familiar interval case,
and which has close connections to trigonometric polynomials.
x1. Introduction
Bernstein­B'ezier (BB­) polynomials defined on an interval are useful tools
for constructing piecewise functional and parametric curves. They play an
important role in CAGD, data fitting and interpolation, and elsewhere. The
purpose of this paper is to develop an analogous theory where the domain of
the polynomials is a circular arc rather than an interval. In addition to their
intrinsic interest, the circular BB­polynomials studied here are also useful for
describing the behavior of spherical BB­polynomials [1] on the circular arcs
making up the edges of spherical triangles.
The paper is organized as follows. In Sect. 2 we introduce a kind of cir­
cular barycentric coordinate which is the basis for our developments. These
are used in Sect. 3 to define circular BB­polynomials. Several basic properties
of BB­polynomials are developed in this section, including a de Casteljau al­
gorithm, subdivision, smoothness conditions for joining BB­polynomials, and